Decimal to Fraction Converter
Convert decimals to simplified fractions and mixed numbers. Shows the continued fraction expansion and multiple equivalent representations.
Simplified Fraction
3/4
Representations
Fraction
3/4
Numerator × Denominator
3 ÷ 4
GCD used
25
Continued Fraction Expansion
[0; 1; 3]
= 0 + 1/(1 + 1/(3))
Convergents (best rational approximations)
| Fraction | Decimal |
|---|---|
| 0/1 | 0.000000 |
| 1/1 | 1.000000 |
| 3/4 | 0.7500000 |
How to Use Decimal to Fraction Converter
- 1Enter a decimal number (e.g. 0.666, 1.75, 3.14159).
- 2See the equivalent simplified fraction and mixed number.
- 3View the continued fraction expansion for the repeating pattern.
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Frequently Asked Questions
How is a decimal converted to a fraction?▾
For a terminating decimal like 0.75: count decimal places (2), multiply numerator and denominator by 10^2 → 75/100, then simplify by GCD(75,100) = 25 → 3/4. For repeating decimals like 0.333…: let x = 0.333…; 10x = 3.333…; 10x − x = 3; 9x = 3; x = 3/9 = 1/3. For mixed repeating like 0.1666…: 100x = 16.666…; 10x = 1.666…; 90x = 15; x = 15/90 = 1/6.
What is the Euclidean algorithm for GCD?▾
The GCD (Greatest Common Divisor) of two integers is found with Euclidean algorithm: GCD(a, b) = GCD(b, a mod b), stopping when b = 0, then GCD = a. Example: GCD(48, 18) → GCD(18, 12) → GCD(12, 6) → GCD(6, 0) = 6. Then simplify: 48/6 = 8, 18/6 = 3 → 8/3. GCD was described by Euclid around 300 BC and remains one of the most efficient algorithms for fraction simplification.
What is a continued fraction?▾
A continued fraction expresses a number as a0 + 1/(a1 + 1/(a2 + ...)) where each aᵢ is a positive integer. Any rational number terminates in finite terms; irrational numbers continue infinitely. Example: 3.245 = 3 + 1/(4 + 1/(12 + ...)). π ≈ [3; 7, 15, 1, 292, ...]. √2 = [1; 2, 2, 2, ...] (periodic). Continued fractions give best rational approximations — [3; 7] = 22/7 is the classic approximation to π. The algorithm: a₀ = floor(x), then take x₁ = 1/(x − a₀), a₁ = floor(x₁), repeat.
What is a mixed number?▾
A mixed number combines a whole part and a proper fraction: 1¾, 2⅓. Convert improper fraction to mixed number: divide numerator by denominator, quotient = whole part, remainder = new numerator. Example: 7/4 → 7 ÷ 4 = 1 remainder 3 → 1¾. Convert back: multiply whole part by denominator, add numerator: 1×4 + 3 = 7 → 7/4. Mixed numbers are easier to visualize (1¾ cups of flour) while improper fractions are easier to compute with.
Why do some decimals give very large fractions?▾
Irrational numbers (π, √2, e) cannot be expressed as exact fractions — only approximations. Many rational decimals with many decimal places have large numerators/denominators before simplification. For example, 0.142857 ≈ 142857/1000000, but it simplifies to 1/7 exactly. The algorithm uses a precision cutoff — very long decimals may hit floating-point limits. For best results, enter exact decimals (e.g., 0.333 for 1/3 approximation) or use fewer decimal places.